A 15 kg, 1m wide door which has frictionless hinges is closed but unlocked. A 400 g ball hits the exact middle of the door at a velocity of 35 m/s and bounces off elastically, thereby causing the door to slowly swing open. How long in seconds does it take for the door to fully open (rotate 90 degrees)?

6.58

To determine how long it takes for the door to fully open, we can analyze the conservation of angular momentum.

Angular momentum is the product of the moment of inertia and the angular velocity of an object. In this case, the moment of inertia of the door is required to calculate the angular momentum.

The moment of inertia of a flat object like a door rotating about an axis perpendicular to its plane can be calculated using the formula:

I = (1/3) * m * r^2

where I is the moment of inertia, m is the mass of the object, and r is the distance from the rotation axis to the object's center of mass. In this case, the mass of the door is 15 kg, and assuming the center of mass is at the center of the door, the distance r would be 0.5 m.

Substituting these values into the formula:

I = (1/3) * 15 kg * (0.5 m)^2

I = 1.25 kg·m²

The initial angular momentum of the ball-door system is the product of the moment of inertia and the initial angular velocity, which is zero since the door starts at rest. Therefore, the initial angular momentum is also zero.

During the collision, the ball imparts some angular momentum to the door. According to the conservation of angular momentum, the total angular momentum before and after the collision remains constant.

The final angular momentum is given by the product of the moment of inertia and the final angular velocity. Since the ball bounces off elastically, it reverses its direction, causing the door to start rotating in the opposite direction. Therefore, the final angular velocity of the door is negative.

The final angular momentum can be calculated using the equation:

L_final = I * ω_final

where L_final is the final angular momentum, I is the moment of inertia, and ω_final is the final angular velocity.

Since the initial angular momentum is zero and the final angular momentum is just the product of the moment of inertia and the final angular velocity, we can write:

0 = I * ω_final

Solving for ω_final:

ω_final = 0 rad/s

The door starts from rest, so it needs to accelerate until it reaches the final angular velocity of 0 rad/s.

To calculate the time it takes for the door to fully open (rotate 90 degrees), we can use the equation for angular displacement:

θ = ω_initial * t + (1/2) * α * t^2

where θ is the angular displacement, ω_initial is the initial angular velocity, α is the angular acceleration, and t is the time.

In this case, the angular displacement is 90 degrees, or π/2 radians. The initial angular velocity is zero, and the final angular velocity is also zero. We're looking for the time, so we can rearrange the equation:

π/2 = (1/2) * α * t^2

Since the angular velocity is zero, α is the angular acceleration.

The angular acceleration α can be calculated using the torque equation:

τ = I * α

where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

In this case, since there are no external torques acting on the system, the only torque is caused by the ball colliding with the door and creating an angular impulse.

The angular impulse is given by:

J = ΔL = L_final - L_initial

where J is the angular impulse, ΔL is the change in angular momentum, L_final is the final angular momentum, and L_initial is the initial angular momentum.

Since the initial angular momentum is zero, we can simplify the equation to:

J = L_final

The angular impulse can be calculated using the equation:

J = r * m * v

where J is the angular impulse, r is the distance from the rotation axis to the point of impact (which is half the width of the door, 0.5 m), m is the mass of the ball (0.4 kg), and v is the velocity of the ball (35 m/s).

Substituting the values:

J = 0.5 m * 0.4 kg * 35 m/s

J = 7 N·s·m

Since the angular impulse is equal to the torque multiplied by the time, we can write:

J = τ * t

Solving for τ:

τ = J / t

The torque τ can be calculated using the equation:

τ = I * α

Substituting the values:

J / t = (1.25 kg·m²) * α

Solving for α:

α = J / (t * I)

Substituting the given values:

α = (7 N·s·m) / (t * 1.25 kg·m²)

Now we can substitute the value of α in the equation for angular displacement:

π/2 = (1/2) * [ (7 N·s·m) / (t * 1.25 kg·m²) ] * t^2

Simplifying:

π/2 = (7 N·m·s / (2 * 1.25 kg)) * t

π/2 = (14/5) * t

Solving for t:

t = (π/2) / (14/5)

t = (5 * π) / 28

t ≈ 0.56 seconds

Therefore, it takes approximately 0.56 seconds for the door to fully open (rotate 90 degrees).